Atividade - 01

Data

The objetive of this homework is to present some stylized facts in Financial Data. First, we colect data from the closing prices of BBAS3 and IBOV:

DADOS_BR <- read_excel("DADOS_BR_SB.xlsx")
#View(DADOS_BR)

DF = data.frame(DADOS_BR$Date, DADOS_BR$BBAS3_SB, DADOS_BR$IBOVC_SB) |> drop_na()

# Banco do Brasil SA (BBAS3)
BBAS.ts = xts(as.numeric(DF$DADOS_BR.BBAS3_SB), order.by=as.Date(DF$DADOS_BR.Date))

# Bovespa Index (IBOV)
IBOV.ts = xts(as.numeric(DF$DADOS_BR.IBOVC_SB), order.by=as.Date(DF$DADOS_BR.Date))

Ploting the closing price

BBAS3

plot(BBAS.ts, main = "BBAS3 Closing Price", ylab = "Price in R$")

IBOV

plot(IBOV.ts, main = "BOVESPA Closing Price", ylab = "Index")

Generating and Ploting Simple Returns

The price series are not stationary so we will use simple returns. Defined by:

\[ R_t = \cfrac{P_t-P_{t-1}}{P_{t-1}} \]

BBAS3

# BBAS3
BBAS.rs.ts = xts(matrix(NA, nrow = 5880, ncol = 1), order.by=as.Date(DF$DADOS_BR.Date))
for (i in 2:5880) {
    BBAS.rs.ts[[i]] = ((BBAS.ts[[i]]-BBAS.ts[[i-1]])/BBAS.ts[[i-1]])
}
plot(BBAS.rs.ts, main = "BBAS3 Simple Return")

IBOV

# IBOV
IBOV.rs.ts = xts(matrix(NA, nrow = 5880, ncol = 1), order.by=as.Date(DF$DADOS_BR.Date))
for (i in 2:5880) {
    IBOV.rs.ts[[i]] = ((IBOV.ts[[i]]-IBOV.ts[[i-1]])/IBOV.ts[[i-1]])
}

plot(IBOV.rs.ts, main = "BOVESPA Simple Return")

Generating and Ploting Compounded Returns

Or, we may use compounded returns. Defined by:

\[ Y_t = ln(P_t)-ln(P_{t-1}) \]

BBAS3

# BBAS3
BBAS.rc.ts = xts(matrix(NA, nrow = 5880, ncol = 1), order.by=as.Date(DF$DADOS_BR.Date))
for (i in 2:5880) {
    BBAS.rc.ts[[i]] = log(BBAS.ts[[i]]) - log(BBAS.ts[[i-1]])
}
plot(BBAS.rc.ts, main = "BBAS3 Compounded Return")

IBOV

# IBOV
IBOV.rc.ts = xts(matrix(NA, nrow = 5880, ncol = 1), order.by=as.Date(DF$DADOS_BR.Date))
for (i in 2:5880) {
    IBOV.rc.ts[[i]] = log(IBOV.ts[[i]]) - log(IBOV.ts[[i-1]])
}

plot(IBOV.rc.ts, main = "BOVESPA Compounded Return")

Summary statistics

We have some summary statistics for the assets. We also run the Test Jarque-Bera.

BBAS3

This indicates that the test statistic is 8328.2, with a p-value of < 2.2e-16. We would reject the null hypothesis that the data is normally distributed in this circumstance.

basicStats(BBAS.rc.ts) |>   
  kbl(caption = "EstatĂ­sticas Descritivas para BBAS") |>
  kable_classic(full_width = F)
EstatĂ­sticas Descritivas para BBAS
x
nobs 5880.000000
NAs 1.000000
Minimum -0.237891
Maximum 0.188256
  1. Quartile
-0.013513
  1. Quartile
0.014448
Mean 0.000656
Median 0.000205
Sum 3.853862
SE Mean 0.000343
LCL Mean -0.000017
UCL Mean 0.001328
Variance 0.000692
Stdev 0.026308
Skewness -0.068001
Kurtosis 5.826231 
jarque.bera.test(BBAS.rc.ts[2:5880])
## 
##  Jarque Bera Test
## 
## data:  BBAS.rc.ts[2:5880]
## X-squared = 8328.2, df = 2, p-value < 2.2e-16

IBOV

This indicates that the test statistic is 13427, with a p-value of < 2.2e-16. We would reject the null hypothesis that the data is normally distributed in this circumstance.

basicStats(IBOV.rc.ts) |>   
  kbl(caption = "EstatĂ­sticas Descritivas para IBOV") |>
  kable_classic(full_width = F)
EstatĂ­sticas Descritivas para IBOV
x
nobs 5880.000000
NAs 1.000000
Minimum -0.159938
Maximum 0.136794
  1. Quartile
-0.008567
  1. Quartile
0.009668
Mean 0.000296
Median 0.000711
Sum 1.740984
SE Mean 0.000224
LCL Mean -0.000143
UCL Mean 0.000735
Variance 0.000294
Stdev 0.017159
Skewness -0.396359
Kurtosis 7.357444 
jarque.bera.test(IBOV.rc.ts[2:5880])
## 
##  Jarque Bera Test
## 
## data:  IBOV.rc.ts[2:5880]
## X-squared = 13427, df = 2, p-value < 2.2e-16

Histogram of Returns

Indeed, the data`s distribution has heavier tails and a taller peak. In red line, we have the data and the blue line is a Normal Distribution.

BBAS3

# BBAS3
m.bbas = mean(BBAS.rc.ts, na.rm=TRUE)
std.bbas = sqrt(var(BBAS.rc.ts, na.rm=TRUE))



hist(BBAS.rc.ts, breaks=100, prob=T, 
     xlab="BBAS3", ylim=c(0, 25), xlim=c(-0.1,0.1),
     main="BBAS3 Returns")
lines(density(BBAS.rc.ts[2:5880]), col = "red")   
curve(dnorm(x, mean=m.bbas, sd=std.bbas), 
      col="darkblue", lwd=2, add=TRUE, yaxt="n")

IBOV

#IBOV
m.ibov = mean(IBOV.rc.ts, na.rm=TRUE)
std.ibov = sqrt(var(IBOV.rc.ts, na.rm=TRUE))

hist(IBOV.rc.ts, breaks=100, prob=TRUE, 
     xlab="IBOV", ylim=c(0, 30), xlim=c(-0.05,0.05),
     main="BOVESPA Returns")
lines(density(IBOV.rc.ts[2:5880]), col = "red")   
curve(dnorm(x, mean=m.ibov, sd=std.ibov), 
      col="darkblue", lwd=2, add=TRUE, yaxt="n")

ACF and PACF of Returns

We can see in both process that ACF has several autocorrelations that are significantly non-zero. Therefore, the time series is non-random. In addition, the PACF does not have a strong correlation of the initial lags that suggests white noise.

BBAS3

# BBAS3
par(mfcol=c(1,2))
acf(BBAS.rc.ts, lag.max=20,main='ACF BBAS3 Returns',  na.action = na.pass)
pacf(BBAS.rc.ts, lag.max=20,main='PACF BBAS3 Returns',  na.action = na.pass)

IBOV

# IBOV
par(mfcol=c(1,2))
acf(IBOV.rc.ts, lag.max=20,main='ACF IBOV Returns',  na.action = na.pass)
pacf(IBOV.rc.ts, lag.max=20,main='PACF IBOV Returns',  na.action = na.pass)

ACF and PACF of Returns Squared

In this case, all correlations are strong for all lags in the ACF and the PACF has out of range correlations that suggest a non-stationary Random-Walk process.

BBAS3

# BBAS3
par(mfcol=c(1,2))
acf((BBAS.rc.ts)^2, lag.max=20,main='ACF BBAS3 Returns Squared',  na.action = na.pass)
pacf((BBAS.rc.ts)^2, lag.max=20,main='PACF BBAS3 Returns Squared',  na.action = na.pass)

IBOV

# IBOV
par(mfcol=c(1,2))
acf((IBOV.rc.ts)^2, lag.max=20,main='ACF IBOV Returns Squared',  na.action = na.pass)
pacf((IBOV.rc.ts)^2, lag.max=20,main='PACF IBOV Returns Squared',  na.action = na.pass)

Squared Returns Plots

For cluster volatility, the high and low volatility moments ocorrs in same moments.

BBAS3

# BBAS3
par(mfcol=c(2,1))
plot(BBAS.rc.ts, main = "BBAS3 Returns")
plot((BBAS.rc.ts)^2, main = "BBAS3 Squared Returns")

IBOV

# IBOV
par(mfcol=c(2,1))
plot(IBOV.rc.ts, main = "IBOV Returns")
plot((IBOV.rc.ts)^2, main = "IBOV Squared Returns")

Correlation Plot

par(mfrow=c(1,2))
# Correlation Returns
plot(as.numeric(BBAS.rc.ts), as.numeric(IBOV.rc.ts), pch = 19, col = "blue", xlab = "BBAS3", ylab = "IBOV", main = "Correlation")
abline(lm(as.numeric(BBAS.rc.ts) ~ as.numeric(IBOV.rc.ts)), col = "red", lwd = 3)

# Correlation Returns Squared
plot(as.numeric((BBAS.rc.ts)^2), as.numeric((IBOV.rc.ts)^2), pch = 19, col = "blue", xlab = "BBAS3 SQ", ylab = "IBOV SQ", main = "Correlation")
abline(lm(as.numeric((BBAS.rc.ts)^2) ~ as.numeric((IBOV.rc.ts)^2)), col = "red", lwd = 3)

Leverage Effects

For leverage effects, price movements are adjusted with volatility. If the asset price is falling, there is an increase in the firm’s leverage and, in general, also an increase in uncertainty, which increases the volatility.

BBAS3

par(mfcol=c(2,1))
plot(BBAS.ts, main = "BBAS3")
plot((BBAS.rc.ts)^2, main = "BBAS3 Squared Returns")

IBOV

# IBOV
par(mfcol=c(2,1))
plot(IBOV.ts, main = "IBOV")
plot((IBOV.rc.ts)^2, main = "IBOV Squared Returns")

Co-movimentos de volatilidade

When market shocks occur in 2008, 2017 and 2020, we have that the volatilities follow for the two assets in their return.

par(mfcol=c(2,1))
plot(BBAS.rc.ts, main = "BBAS3 Returns")

plot(IBOV.rc.ts, main = "IBOV Returns")